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Answer :
To find [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex], we can follow these steps:
1. Understand the Division of Fractions: Dividing by a fraction is the same as multiplying by its reciprocal. So, [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] becomes [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex].
2. Multiply the Fractions: When multiplying two fractions, you multiply the numerators together and the denominators together:
[tex]\[
\text{Numerator: } 14 \times 5 = 70
\][/tex]
[tex]\[
\text{Denominator: } 15 \times 7 = 105
\][/tex]
This gives us [tex]\(\frac{70}{105}\)[/tex].
3. Simplify the Fraction: To simplify [tex]\(\frac{70}{105}\)[/tex], find the greatest common divisor (GCD) of 70 and 105:
- Both 70 and 105 are divisible by 35.
- Divide both the numerator and the denominator by 35:
[tex]\[
\frac{70 \div 35}{105 \div 35} = \frac{2}{3}
\][/tex]
So, [tex]\(\frac{14}{15} \div \frac{7}{5} = \frac{2}{3}\)[/tex].
Let's connect this reasoning to the given statements:
- [A] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]: This statement is correct because it follows the process of multiplying by the reciprocal. First, multiply [tex]\(\frac{14}{15}\)[/tex] by 5, then by [tex]\(\frac{1}{7}\)[/tex], simplifying the expression as [tex]\(\frac{2}{3}\)[/tex].
- [C] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex]: This statement is incorrect since it doesn't involve the correct steps of multiplying by the reciprocal of [tex]\(\frac{7}{5}\)[/tex].
- [E] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7 and then dividing by 5: This statement is incorrect as it would not follow the correct operation of division by a fraction through multiplication with the reciprocal.
Thus, the correct statements that show the proper reasoning are [A] and none of the others.
1. Understand the Division of Fractions: Dividing by a fraction is the same as multiplying by its reciprocal. So, [tex]\(\frac{14}{15} \div \frac{7}{5}\)[/tex] becomes [tex]\(\frac{14}{15} \times \frac{5}{7}\)[/tex].
2. Multiply the Fractions: When multiplying two fractions, you multiply the numerators together and the denominators together:
[tex]\[
\text{Numerator: } 14 \times 5 = 70
\][/tex]
[tex]\[
\text{Denominator: } 15 \times 7 = 105
\][/tex]
This gives us [tex]\(\frac{70}{105}\)[/tex].
3. Simplify the Fraction: To simplify [tex]\(\frac{70}{105}\)[/tex], find the greatest common divisor (GCD) of 70 and 105:
- Both 70 and 105 are divisible by 35.
- Divide both the numerator and the denominator by 35:
[tex]\[
\frac{70 \div 35}{105 \div 35} = \frac{2}{3}
\][/tex]
So, [tex]\(\frac{14}{15} \div \frac{7}{5} = \frac{2}{3}\)[/tex].
Let's connect this reasoning to the given statements:
- [A] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 5 and then by [tex]\(\frac{1}{7}\)[/tex]: This statement is correct because it follows the process of multiplying by the reciprocal. First, multiply [tex]\(\frac{14}{15}\)[/tex] by 5, then by [tex]\(\frac{1}{7}\)[/tex], simplifying the expression as [tex]\(\frac{2}{3}\)[/tex].
- [C] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7, and then multiplying by [tex]\(\frac{1}{5}\)[/tex]: This statement is incorrect since it doesn't involve the correct steps of multiplying by the reciprocal of [tex]\(\frac{7}{5}\)[/tex].
- [E] Multiplying [tex]\(\frac{14}{15}\)[/tex] by 7 and then dividing by 5: This statement is incorrect as it would not follow the correct operation of division by a fraction through multiplication with the reciprocal.
Thus, the correct statements that show the proper reasoning are [A] and none of the others.
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